Because then people would game it.
Playing devil's advocate, gaming search engines used to be about keywords then it became more about links. SEOs nowadays talk about "EAT", expertise, authority, and trustworthiness. It's still possible for marketers to mimic quality signals but it's less easy.
It would be much easier if the author of webpages were to declare themselves, sort of a Pagerank but applied on a person/topic level. It could/can be done with structured data. This of course has privacy implications but it keeps the publishers as transparent to the search engine as possible. Verification of sorts would be needed so other people don't use your authorship details on their own site. It's the kind of social fabric/data Google could've perhaps used Google+ for.
Slightly more on topic, marketers used to be able to do accurate keyword research with Google, and also see the keywords that people searched for when landing on their site, which is quite useful for figuring out their intent. Google keeps all that data to themselves now.
I find their "concerns" about "use of private data" a bit laughable, considering they used your personal search history to try and target you on sites you visit across the web via ads. They don't mind retaining data about users, they just don't like having to share it with anyone.
This was actually done back when Google+ first was launched — people could attach their authorship and it would be displayed on the search page, and you would also see a list of “experts” for particular categories on the right hand side when you searched for things like “python” or “poker”.
I believe it disappeared with the rest of Google+, because no one actually cared.
...with slightly more effectiveness allowing better competition with google itself.
You can do search. You can provide content. Doing both became rent seeking. And damn does google do rent-seeking very effectively.
The only difference would be that you don't need to do industrial espionage on google to see what works first, so the cost of the seo would drop.